Global and local conservation of mass, momentum and kinetic energy in the simulation of compressible flow

The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions are sought for the local and global conservation of primary (mass and momentum) and secondary (kinetic energy) invariants on Cartesian meshes. The analysis, based on a matrix approach, shows that sharp criteria for global and local conservation can be obtained and that in many cases these two concepts are equivalent. Explicit numerical fluxes are derived in all finite-difference formulations for which global conservation is guaranteed, even for non-uniform Cartesian meshes. The treatment reveals also an intimate relation between conservative finite-difference formulations and cell-centered finite-volume type approaches. This analogy suggests the design of wider classes of finite-difference discretizations locally preserving primary and secondary invariants

Parabolic interface reconstruction for 2D volume of fluid methods

For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young–Laplace jump condition. We show that traditional geometric volume of fluid (VOF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems. Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabolic LVIRA and MOF methods, PLVIRA and PMOF, respectively. We show that a Lagrangian remapping method is sufficiently accurate for the advection of such a parabolic interface. It is numerically demonstrated that the newly proposed PPIC methods result in an increase of reconstruction accuracy by one order, convergence of the interface curvature in time-dependent advection problems and Weber number independent convergence of a droplet translation problem, where the advection method is coupled to a two-phase Navier–Stokes solver. The PLVIRA method is applied to the simulation of a 2D rising bubble, which shows good agreement to a reference solution.</p

Incompressible viscous flow near the leading edge of a flat plate admitting slip

The shear stress at the leading edge, calculated on basis of the Navier-Stokes equations and the no-slip boundary condition, approaches infinity. However, taking into account the mean free path of the molecules, which implies admitting a certain slip, the shear stress becomes inversely proportional to the square root of the Knudsen number κ if κ→0. κ is defined as the ratio between the mean free path and the viscous length. The new boundary condition modifies the shear stress only within the Knudsen region of which the size is of the order of 3 to 4 times the mean free path.

Branching of the Falkner-Skan solutions for λ < 0

The Falkner-Skan equation f'" + ff" + λ(1 - f'^2) = 0, f(0) = f'(0) = 0, is discussed for λ < 0. Two types of problems, one with f'(∞) = 1 and another with f'(∞) = -1, are considered. For λ = 0- a close relation between these two types is found. For λ < -1 both types of problem allow multiple solutions which may be distinguished by an integer N denoting the number of zeros of f' - 1. The numerical results indicate that the solution branches with f'(∞) = 1 and those with f'(∞) = -1 tend towards a common limit curve as N increases indefinitely. Finally a periodic solution, existing for λ < -1, is presented.

Eulerian modeling of inertial and diffusional aerosol deposition in bent pipes

This paper presents a sectional Eulerian aerosol model for size-dependent droplet deposition at walls of the domain, driven by both diffusion and inertia. The model is based on the internally mixed assumption and employs the formulation for compressible aerosols. It is validated in a bent pipe geometry against models and experimental and numerical data from literature. Good agreement is found in both the diffusion and inertial deposition regimes. To improve the overprediction of inertial deposition by a boundary treatment that adopts zero-gradient droplet wall velocity, we use a corrected wall velocity, based on an analytical solution of the droplet motion near the wall. In the bent pipe setting the corrected wall velocity is found to reduce the overprediction of deposition and is less sensitive to grid refinement. We also show that refining the computational mesh near the pipe wall improves the predicted deposition efficiency, significantly. Finally, we present a parameter study varying the Reynolds number and the bend curvature. The deposition efficiency curve is recorded for droplet diameters ranging from the nanometer scale to beyond the micrometer scale, which is a unique contribution of this paper. The complete size range is simulated in only one simulation, due to the sectional approach. In the diffusion-dominated regime an increase in Reynolds number leads to a gradual enhancement of deposition. In the inertial regime, where droplet drift dominates deposition, a much stronger dependence on the Reynolds number is found. The dependence of the deposition on the bend curvature is less pronounced. The results shown in this paper establish the role of Eulerian simulation in predicting deposition of aerosol droplets and are useful for understanding size-dependent aerosol deposition in other more complex confined geometries